Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| isoclinism | isoclinic class of #5997 |
| automorphism | ? |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C7 |
No aliases for this group.
| derived, Sylow | C72 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | #5997 |
| commutator, hypocenter | C72 | C16 |
| Fitting | C4×C72 | C4 |
| socle, prime radical | C2×C72 | C8 |
| cosocle | C7⋊(C7⋊C8) | C2 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C4 | |
26 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^4,7^2|a^2=b,b^2=c,c^2=d,e^a=e^3f^5,f^a=e^5f^4,e^b=e^6,f^b=f^664 conjugacy classes — view the full character table.
| maximal subgroup of | #13096178, #13096187, #13096188, … |
| maximal normal subgroup of | #13096178, #13096187, #13096188, … |
| maximal quotient of | #13096032, C2×#11716027 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 47);